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Counterexample · presentation boundary

QA-BK-N01

Endpoint liftability is weaker than factorwise liftability

Exact statement

When ε(t_c) is nonzero, the element t_c² is liftable although the displayed factorization (t_c)(t_c) has two individually unliftable factors; regrouping changes the admissible atom grammar without being a Hurwitz move.

StatusExact minimal witness proved
External reviewNo documented external or specialist review of this FPRD deduction is recorded.

Context

A product can return the sheet phase to zero even though its displayed factors leave the liftable locus after each step.

Definitions

  • A factorization is factorwise liftable when every displayed factor has zero boundary-phase receipt.

Hypotheses and scope

  • ε(t_c)≠0.

Proof or evidence

Homomorphy gives ε(t_c²)=2ε(t_c)=0 over F₂ while each singleton retains ε(t_c)≠0.

Packetization boundary →

Verification notes

The thesis supplies the squared-twist and marked-point-to-boundary conventions; the FPRD question is how an algorithm represents packetization.

Limitations

  • No claim is made that regrouping preserves Lefschetz-factor counts or strict Hurwitz equivalence.

Open work

Treat packetization as an explicit allowed operation in any algorithm comparing liftable positive factorizations.